Graded D-branes and skew-categories

TL;DR

This paper develops a formalism linking graded open 2D topological field theories with equivariant cyclic categories using skew categories, and applies it to graded D-branes in Landau-Ginzburg models.

Contribution

It introduces a new approach to describe graded open 2D TFTs via skew categories and applies this to Landau-Ginzburg models, proposing a conjecture for the Serre functor.

Findings

Formalism translating graded TFTs to equivariant cyclic categories

Application to graded D-branes in Landau-Ginzburg models

Conjecture for the Serre functor on graded matrix factorizations

Abstract

I describe extended gradings of open topological field theories in two dimensions in terms of skew categories, proving a result which alows one to translate between the formalism of graded open 2d TFTs and equivariant cyclic categories. As an application of this formalism, I describe the open 2d TFT of graded D-branes in Landau-Ginzburg models in terms of an equivariant cyclic structure on the triangulated category of `graded matrix factorizations' introduced by Orlov. This leads to a specific conjecture for the Serre functor on the latter, which generalizes results known from the minimal and Calabi-Yau cases. I also give a description of the open 2d TFT of such models which manifestly displays the full grading induced by the vector-axial R-symmetry group.